I realized this morning that I had overlooked some of my algebra:
So, moving through a few more steps I get to:
I'm not sure if it's any better, but it's something ...
I am trying to show that the square of any Fibonacci Number differs by one from the product of its neighbors:
My best attempt is to substitute Binet's Formula:
and simplify, but keep getting buggered near the end. I get down to:
and can go no further due to the unequal bases. I know (because even values of "n" yield -1) that:
which would make the final formula:
but I seem to be missing something. I could swear I've seen this done, but can't find it anywhere. Any pointers as to what I'm missing?
Wikipedia has a nice proof of this theorem (Cassini's identity):
Cassini and Catalan identities - Wikipedia, the free encyclopedia
That is nice - I was hoping for something algebraic, but this is about as straightforward as I could ask for. I have two concerns with this solution though:
1. Using a matrix looks an awful lot like we simply restated the problem and called it solved
2. While creating the matrix using with an odd value of n works well, won't using an even value of n yield a determinant of which will always be positive?
It looks like I'll have to brush up on matrices so I can explain it. Wikipedia to the rescue once again!
Yes on both counts. My "concerns" were rather poorly worded - what I meant to convey is that I'm worried my unfamiliarity with matrices leaves me unprepared to answer what should be a couple of simple questions, not that there was a problem with the proof. Hence my need to "brush up."